5.2 Multi-qubit gates (CNOT, SWAP)

Multi-qubit gates are the building blocks of quantum circuits, enabling complex operations on multiple qubits simultaneously. These gates, like CNOT and SWAP, create entanglement and perform controlled operations, essential for quantum algorithms and computations.

Combining multi-qubit gates with single-qubit gates forms a universal set for quantum computing. This allows for the construction of quantum circuits that can implement various algorithms, optimize qubit arrangements, and manipulate entangled states for quantum speedup.

Multi-Qubit Gates

Operation of CNOT gate

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• CNOT gate performs a controlled operation on two qubits consisting of a control qubit and a target qubit
  • Flips the target qubit (applies a NOT operation) if the control qubit is in the state $|1\rangle$
  • Leaves the target qubit unchanged if the control qubit is in the state $|0\rangle$
• Creates entanglement between two qubits when applied to a superposition state
  • Applying a CNOT gate to the state $\frac{1}{\sqrt{2}}(|00\rangle + |10\rangle)$ results in the entangled state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ (Bell state)
• Represented by the matrix: $\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{bmatrix}$

Application of SWAP gate

• SWAP gate exchanges the states of two qubits by applying a sequence of three CNOT gates
  • Sequence: CNOT(q1, q2), CNOT(q2, q1), CNOT(q1, q2), where q1 and q2 are the two qubits involved in the swap
• Rearranges the order of qubits in a quantum circuit, which can be used to optimize circuits by reducing the number of required gates
• Represented by the matrix: $\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$

Role of multi-qubit gates

• Essential for creating entanglement, a key resource in quantum computing that enables certain quantum algorithms to outperform classical algorithms (quantum speedup)
• Implement various quantum algorithms such as Quantum Fourier Transform (QFT), Grover's search algorithm, and Shor's factoring algorithm
  • These algorithms rely on the ability to create and manipulate entangled states using multi-qubit gates
• Combined with single-qubit gates, form a universal set of quantum gates capable of performing any quantum computation

Construction of quantum circuits

• Built using a combination of single-qubit gates (Pauli gates, Hadamard gate) and multi-qubit gates (CNOT, SWAP)
  • Single-qubit gates manipulate individual qubits
  • Multi-qubit gates create entanglement and perform controlled operations
• Designed to implement specific quantum algorithms or prepare desired quantum states
  • Represented using quantum circuit diagrams showing the sequence of gates applied to the qubits
• Optimized to minimize the number of gates and circuit depth (number of gate layers)
  • Techniques such as gate decomposition and circuit rewriting can be used to optimize circuits